Skip to main content

Full text of "Mathematical And Physical Papers - Iii"

See other formats

ON  THE  MOTION OF PENDULUMS.                            17
systems ; T: T',T"... corresponding times, such for example as the times of oscillation from rest to rest. Let x, y, z be measured from similarly situated origins, and in corresponding directions, and t from corresponding epochs, such for example as the commencements of oscillations when the systems are beginning to move from a given side of the mean position.
The form of equations (2), (3) shews that the equations being satisfied for one system will be satisfied for all the systems provided
n        ULU    pux u oc v oc w, x oc y oc z, and p oc  oc ~- .
3?                    6
The variations x oc y oc z merely signify that we must compare similarly situated points in inferring from the circumstance that (2), (3) are satisfied for one system that they will be satisfied for all the systems. If c, c', c"... be the maximum excursions of similarly situated points of the fluids
f*                                                                                                             *
u oc -yi > x x a, t oc Ty
and the sole condition to be satisfied, in addition to that of geometrical similarity, in order that the systems should be dynamically similar, becomes
^oc^or oc/*'..........................(6).
This condition being satisfied, similar motions will take place in the different systems, and we shall have
pac                                           /rrx
It follows from the equations (4), (5), and the other equations which might be written down from symmetry, that the pressures such as P1? T3 vary in the same manner as p, whence it appears from (7) that the resultant or resultants of the pressures of the fluids on the solids, acting along similarly situated lines, which vary asjpa2, vary as pa? and cT~2 conjointly. In other words, these resultants in two similar systems are to one another in a ratio compounded of the ratio of the masses of fluid displaced, and of the ratio of the maximum accelerating effective forces belonging to similarly situated points in the solids.
6.    In order that two systems should be similar in which the fluids are confined by envelopes that are sufficiently narrow to s.   in.                                                                         2